OCIV. Orbital Concepts and Their Applications in Organic Chemistry. Klaus Müller. Script ETH Zürich, Spring Semester 2010. Chapter 3. sCharacter balancing for central atom hAO’s Ligand geometries around central atoms.
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Orbital Concepts
and
Their Applications in Organic Chemistry
Klaus Müller
Script
ETH Zürich, Spring Semester 2010
Chapter 3
sCharacter balancing for central atom hAO’s
Ligand geometries around central atoms
In principle, sp3 hAO‘s is a good starting point for a tetrahedrally coordinated
central atom, even if the ligand geometry deviates somewhat from an ideal
tetrahedral geometry.
There may be various reasons for he observation of ligand geometries that
deviate markedly from ideal tetrahedral coordination:
 steric interactions between the ligands
 electronic interactions between the ligands other than steric
 nontetrahedral valence angle(s) when ligands are involved in rings
 different electronegativities of the ligands
electropositive ligand
the sLMO is polarized
towards the central atom;
the coefficient cs at the
central atom for this sLMO
is comparatively large;
electronic energy lowering
by shifting scharacter from
lowamplitude hAO domain to
highamplitude hAO domain
hence, any change in
scharacter of the hAO
of the central atom will be
strongly felt by this sLMO
electronegative ligand
the sLMO is polarized
towards the ligand;
the coefficient cs at the
central atom for this sLMO
is comparatively small;
Bent‘s rule:
a central atom increases the
scharacter of its hAOs oriented
towards electropositive ligands
at the expense of scharacter
in hAO‘s towards electronegative
ligands, i.e.,
hence, any change in
scharacter of the hAO
of the central atom will be
little felt by this sLMO
a central atom does not waste scharacter
towards electronegative ligands
114.7°
115.9°
106.7°
103.8°
102.4°
hypothetical ligand
of zero electronegativity
sLMO polarized towards central atom;
any gain in scharacter of center hAO
results in marked energy gain of sLMO
sLMO in the extreme
becomes a lonepair orbital
sLMO polarized towards electronegative
ligand atom; any change in scharacter of
center hAO has little effect on energy of sLMO
valence electron sextett
108.4°
120.0°
110.5°
hypothetical ligand
of infinite electronegativity
sLMO polarized towards electronegative
ligand atom; any change in scharacter of
center hAO has little effect on energy of sLMO
sLMO in the extreme
becomes an empty hAO;
no scharacter is wasted
for empty orbital: hence,
empty orbital is pure pAO;
→ planar geometry
sLMO polarized towards central atom;
any gain in scharacter of center hAO
results in marked energy gain of sLMO
three sLMO‘s receive all scharacter
in symmetrical case: three sp2hAO‘s;
geometry is trigonal planar
g
f2
x
f1
g
180°
170°
160°
1
150°
1 
2
140°
cp
130°
120°
2
 cs
110°
2
1  cs
100°
90°
hAO pcharacter
50%
60%
70%
80%
80%
100%
hAO scharacter
50%
40%
30%
20%
10%
0%
sp
sp2
sp3
sp5
p
relationships between
s and pcharacter of two equivalent hAO‘s
and the interorbital axis angle g
2
2
f1 = cs.s + cp.px
normalization : cs + cp = 1
2
2
f2 = cs.s + cp. (cosg.px + sing.py)
orthogonality : cs + cp.cosg = 0
2
2
hence : 1  cp + cp.cosg = 0
1
2
pcharacter : cp=
1  cosg
cosg
2
scharacter : cs=
1  cosg
g from pcharacter : cosg=
g from scharacter : cosg=
108.4° ± 2.6°
109.6° ± 1.8°
111.3° ± 2.0°
112.7° ± 0.2°
108.1° ± 2.2°
109.1° ± 2.0°
111.0° ± 1.5°
110.9° ± 0.2°
104.5°
92.1°
90.6°
90.3°
111.7°
98.9°
96.2°
106.6
92.2
87.8
1.714
1.855
1.362
..
..
0.14  0.20 Å
q = 67°
q
q = 3.5° ± 3.7°
q ~ 70°!
Xray of
1benzyl
phosphole
(BZPHOS10)
pyrroles essentially planar
pconjugation pays for
lone pair spx→ p promotion
+
106.7°
109.8°
119.7°
110.9°
107.1°
102.4°
93.3°
98.9°
101.7°
113.0°
q
q
q
q
q = 59.1° ± 3.4°
q = 49.3° ± 3.5°
q = 46.3° ± 6.5°
q = 49.1° ± 5.7°
0.66 ± 0.04 Å
0.51 ± 0.04 Å
0.43 ± 0.06 Å
0.44 ± 0.05 Å
h (pyr.height)
~ 19 kcal/mol
~ 10 kcal/mol
~ 8 kcal/mol
~ 8 kcal/mol
N
inversion barrier
e
hn
IP1
9.9 eV
9.0 eV
8.8 eV
8.7 eV
8.0
11.3
11.3
11.1
pKa(R=H)
7.9

10.3
10.1
pKa(R=CH3)
amine basicity (in H2O)
215.7 kcal/mol
222.7
224.3
225.4 kcal/mol
PA (R=H)
221.5 kcal/mol

227.8
228.8 kcal/mol
PA (R=CH3)
proton affinity in gasphase
DGNinv
~ 8 kcal/mol
1316 kcal/mol
2630 kcal/mol
very slow at RT
very fast at RT
fast at RT
rates (RT)
~ 107 sec
~ 103 – 101 sec
~ 106 – 109 sec
t1/2 (RT)
~ 107 sec
~ 103 – 101 sec
~ 106 – 109 sec
~ 10d– 10y
first diastereomeric cis and trans
Nmethoxy isoxazolidin derivatives
isolated by
K. Müller & A.Eschenmoser,
Helv.Chim.Acta 52, 1823 (1969)
‡
DGNinv
~ 1820 kcal/mol
2528 kcal/mol
>32 kcal/mol
very slow at RT
fast at RT
very slow at RT
first diastereomeric cis and trans
NClaziridine derivatives isolated by
A.Eschenmoser & D. Felix,
Angew. Chem IE 7, 224 (1968)
‡
DGNinv
~ 2832 kcal/mol
~ 2628 kcal/mol
very slow at RT
very slow at RT
kB.T
DG#
.
e
k =
RT
h
t1/2 = ln2 / k
T
k (sec1)
80°C
0°C
25°C
100°C
DG#
5
9.106
6.108
1.109
9.109
kcal/mol
10
2.101
6.104
3.105
1.107
5.105
7.101
6.100
1.104
15
1.1010
20
6.104
2.102
2.101
7.108
25
3.1016
3.106
2.102
6.1022
7.1012
30
8.1010
2.105
t1/2 (sec)
T
80°C
0°C
25°C
100°C
DG#
5
7.108
1.109
5.1010
7.1011
kcal/mol
10
1.105
2.106
6.108
3.102
15
~4h
1.101
1.102
5.105
20
5.101
4.102
~200y
~20min
~108y
3.101
25
~120d
~2d
~8h
30
~1013y
~3000y
~30y
z
z
y
y
y
x
x
x
hAO2 = csss + csp (cosj ( py + px) – sinj pz)
hAO3 = csss + csp (cosj ( py  px) – sinj pz)
2
2
cns + 3 css = 1 (1 valence sAO available, eq 2)
2
2
2
cnpz + 3 csp sin j = 1 (1 valence pzAO, eq 3)
2
2
1
3
3
3
2
2
3 csp cos j = 2 (2 valence px,yAO‘s, eq 4)
2
2
cns + cnpz = 1 (nHAO normalized, eq 1)
1
1
√3
√3
2
2
2
2
2
2
csp =
csp =
2
2
2
cnpz =
css =
cns =
quantitative relationships between hAO s and pcharacters
for a trigonal pyramidal center with a lone pair as a function of ligand geometry
r
r
j
j
r
g
r
d
1 lonepair spn hybridAO
pojnting along zaxis
3 equivalent spm hybridAO‘s
pojnting along axes
of trigonal pyramid
relationship between
outofplane angle j
and valence bond angle g
d2 = 2r2 – 2r2 cos g
d2 = 2r2 – 2r2 cos 120°
r= r cosj
cos2j = (1 – cosg)
hAOn = cnss + cnpzpz
hAO1 = csss + csp (cosj py – sinj pz)
contraints and normalization:
1
1
(from eq 4)
hence:
cos j
1  cosg
2
 3 cosg
2
2
cnpz = 1 – 2 tg j
(from eq 3)
1  cosg
1 + 2 cosg
2
2
cns = 2 tg j
(from eq 1)
1  cosg
 cosg
2
2
css = (1  2 tg j)
(from eq 2)
1  cosg
110
3
2
cosg = 1  cos j
2
100
90
2
2
80
cos j
cos j
70
 2
3
cosq =
60
√
4  3
50
40
30
20
10
0
q
j
g
35
30
25
20
15
10
5
0
5
10
15
20
25
30
35
j
e(p)
e(shAO)
e(sp3)
e(sp2)
e(nhAO)
e(s)
for extreme pyramidalities,
the poor sorbital overlap of
a pAO with the ligand hAO,
results in a destabilization of
the sLMOs
e(sLMO)
in this domain, the sLMO‘s show
a rather flat energy response to the
rehybridization of the hAO‘s at the
central atom; this response is the
weaker the more the sLMO‘s are
polarized towards electronegative
ligands; hence, in this domain, the
energy of the doubly occupied lone
pair orbital dictates the geometry
by pulling as much scharacter as
possible from the hAO‘s involved
in the sLMO‘s
trig
tet
tet
q = tet/2
quantitative relationships between hAO s and pcharacters
for a tetrahedral center with two different sets of ligands
z
z
z
y
y
y
x
x
x
2
2 sin gA/2 = 1 – cos gA
2
2
2 csA + 2 csB = 1
2
2
2
2
2 cpB sin gB/2 = 1
2 cpA sin gA/2 = 1
2
2
2
2
csB + cpB = 1
csA + cpA = 1
1
1
2
cpA = =
2
2 sin gA/2
1  cos gA
 cos gA
1
2
csA = 1  =
1  cos gA
1  cos gA
2
1 – 2 csA
1 + cos gA
2
csB = =
2
2 (1  cos gA)
_
_
_
1 + cos gA
cos gB = = –
+
+
+
1  3 cos gA
2
2
1 – 3 cos gA
cpB = 1 – csB =
2 (1  cos gA)
2
2
cpB  1
cpB  1
2
2
≈ cos gAd
cpB
cpB
+
2
2
√2
√2
3
3
√2
2 equivalent
spm hAO‘s
along ZA axes
A
A
A
A
A
A
gA
B
B
B
gB
2 equivalent
spn hAO‘s
along ZB axes
B
B
B
hAOA1,2 = csA s + cpA (cos gA/2 pz± sin gA/2 py)
gA given; gB = f(gA)
hAOB3,4 = csB s + cpA ( cos gB/2 pz± sin gB/2 px)
contraints and normalization:
(1 svalence AO, eq 1)
(1 pyvalence AO, eq 2)
(1 pxvalence AO, eq 3)
(normalization, eq 4)
(normalization, eq 5)
(from eq 2)
hence:
(from eq 4)
(from eq 1)
(from eq 5)
(from eq 3)
angular deviation: gA = (tet) ± d
cos (gA ± d) = cos gA cos d sin gA sin d
1+ cos gAd
1 + cos gA
1
d
cos gB = = – ≈ =
(1 )
hence:
3
1 – 3 cos gA
2
opening of gA results in closing of gB and vice versa; and
hence:
increase in scharacter in hAOA‘s and decrease in scharacter in HAOB‘s
1 + cos gA
1 + cos gA
cos gB = –
cos gB = –
1 – 3 cos gA
1 – 3 cos gA
Search in Cambridge Structural DatabaseA = {Ctet}
B = {N, O, F, P, S, Cl}
all bonds acyclic at central C
results in 253 Xray structures:
gB
108.3°
gA
113.7°
gB
1 + cos gA
cos gB = –
1 – 3 cos gA
gB = tet
gB = 107.5° ± 3.2°
gAgB scattergram
gA
gA = tet
gA = 115.4° ± 3.0°
Search in Cambridge Structural DatabaseA = {C, N, O, F)
all bonds acyclic at central C
results in 12‘624 Xray structures:
gB
gB
gA
1 + cos gA
cos gB = –
1 – 3 cos gA
gB = tet
gAgB scattergram
gA
gA = tet
109.9° ± 4.1°
108.5° ± 4.0°
108.0° ± 3.4°
113.2° ± 1.5°
110.4° ± 1.2°
108.1° ± 2.0°
107.6° ± 1.1°
115.9° ± 2.0°
110.6° ± 1.3°
110.9° ± 3.5°
109.7° ± 1.9°
no equivalent orthonormal spx hAOs
possible for bond vectors of < 90°
interbond angle
equivalent orthonormal spx hAOs
along bond vectors of 90° interbond
angle would be pure pAOs resulting
in poor sLMOs and unacceptable
angle (180°!) for exocyclic (sp) hAOs
two equivalent spx hAOs (ca. sp5)
defined by equivalent hAOs (ca. sp2)
along exocyclic bond axes;
two equivalent spx hAOs (ca. sp3)
defined by equivalent hAOs along
exocyclic bond axes
sp5 hAO axes deviate ca.20° from
CC bond axes of cyclopropane;
offaxis hAOs result in significantly
bent sLMOs (s*LMOs)
sp3 hAO axes deviate only ca.10°
from CC bond axes of cyclobutane; offaxis hAOs result in only slightly
bent sLMOs (s*LMOs)